Question

How can a monopolist identify the profitmaximizing level of output if it knows its total revenue and total cost curves?

Short answer

A monopolist can identify the profit-maximizing level of output by equating its marginal revenue (MR) and marginal cost (MC) curves. To do this, the monopolist should first find the derivatives of its total revenue (TR) and total cost (TC) curves with respect to output quantity (Q). Then, the profit-maximizing level of output (Q*) is found by solving the equation MR = MC. For example, if the given TR and TC curves are \(TR = 100Q - Q^2\) and \(TC = 20Q + Q^2\), the profit-maximizing output level is 20 units, as \(100 - 2Q = 20 + 2Q\) yields \(Q* = 20\).

Step-by-step solution

Understand the terms

Total Revenue (TR): The total income generated from selling a certain quantity of output. It can be calculated by multiplying the price per unit (P) by the quantity sold (Q), i.e., TR = P * Q. Total Cost (TC): The total cost incurred in producing a certain quantity of output. It includes fixed costs (that do not vary with the level of production) and variable costs (that change with the level of production). Profit (π): The difference between the total revenue and total cost, i.e., π = TR - TC. Marginal Revenue (MR): The additional revenue generated from selling one more unit of output. It can be calculated as the derivative of the total revenue with respect to the quantity (Q), i.e., MR = d(TR)/dQ. Marginal Cost (MC): The additional cost incurred in producing one more unit of output. It can be calculated as the derivative of the total cost with respect to the quantity (Q), i.e., MC = d(TC)/dQ.

Find the marginal revenue and marginal cost

Using the given total revenue and total cost curves, we need to find their respective derivatives with respect to the quantity (Q). This will give us the marginal revenue (MR) and marginal cost (MC) curves. For example, let's assume the following TR and TC curves: Total Revenue: \(TR = 100Q - Q^2\) Total Cost: \(TC = 20Q + Q^2\) Now, find the derivatives of TR and TC with respect to Q: \(MR = d(TR)/dQ = 100 - 2Q\) \(MC = d(TC)/dQ = 20 + 2Q\)

Equate marginal revenue and marginal cost

To find the profit-maximizing level of output (Q*), we need to equate the marginal revenue (MR) with the marginal cost (MC), i.e., MR = MC. In our example: \(100 - 2Q = 20 + 2Q\)

Solve for the profit-maximizing level of output

Now we need to solve the equation to find the profit-maximizing level of output (Q*). \(100 - 2Q = 20 + 2Q\) \(100 - 20 = 2Q + 2Q\) \(80 = 4Q\) \(Q* = 20\)

Interpret the result

The monopolist can maximize its profit by producing and selling 20 units of output. This is the point where the difference between total revenue and total cost is maximized, and consequently, where the marginal revenue is equal to the marginal cost.